Chessboard problems

You don’t have to know chess to solve them.

Created by Inna Shapiro ©2008

Problem 1

Can one cover an 8 x 8 chessboard with 2 x 1 dominoes?The dominoes cannot overlap.

AnswerYes, see the picture below. A step-by-step covering is shown on the Slide Show.

and so on.

Problem 2

Can 2 x 2 dominoes cover an 8 x 8 chess board without the upper right square?The dominoes cannot overlap.

Answer

A chess board without the upper right square has 63 squares on it. Each domino piece covers 2 squares. Dominoes can only cover an even number of squares and cannot cover a chess board without one square.

Problem 3

Can you cover an 8 x 8 chessboard without the upper right and lower left square with 2 x 2 dominoes? The dominoes cannot overlap.

AnswerThe upper left and lower right squares have the same color, for example, black. The chess board on the picture has 32 white squares and 30 black ones. Each domino piece covers one black and one white square. That means we cannot cover this chessboard with dominoes.

Problem 4

Can one cover an 8 x 8 chessboard without the upper right and upper left square with 2 x 2 dominoes?The dominoes cannot overlap.

Answer

Yes, see the picture below. A step-by-step covering is shown on the Slide Show.

and so on.

Problem 5

Can you cover an 8 x 8 chessboard with 3 x 1 rectangles like the one shown on the picture? They must not overlap.

Pattern

Answer

An 8 x 8 chessboard has 64 squares on it. Each rectangle covers 3 squares. They cannot cover the chessboard, because 64 is not divisible by 3.

Problem 6

Can you cover an 8 x 8 chessboard without the upper right square with the pieces shown on the picture?The pieces cannot overlap.

and

Answer

Yes, see the picture below.A step-by-step covering is shown on the Slide Show.

Problem 7

Can you cover an 8 x 8 chessboard without the upper right square with pieces like the one shown on the picture? They may not overlap.

Piece

Answer

Yes, see the picture below.A step-by-step covering is shown on the Slide Show.

Problem 8

Can you cover an 8 x 8 chessboard without the upper right square with 1 x 3 rectangles like the one shown on the picture? They are not allowed to overlap.

Hint

Paint the chessboard in three different colors.

AnswerThere are 64 squares on the colored chessboard – 22 blue, 21 red and 21 green. After removing one red corner square, the chess board contains 22 blue, 20 red and 21 green squares. Each 3 x 1 rectangle always covers three different colors. That means that a chessboard without the upper right square cannot be covered by these rectangles.

Problem 9

Can one cover a 10 x 10 chessboard with pieces like the one shown on the picture?The pieces may not overlap.

HintEach piece can either cover three white squares and a black one (1), or three black squares and a white one (2).

1

2

AnswerA 10 x 10 chessboard has 100 squares. One piece covers 4 squares. That means we need 25 pieces to cover the board. There is an equal number of black and white squares on the chessboard, therefore we need an equal number of pieces of type 1 and 2. But 25 is not divisible by 2. That means the chessboard cannot be covered by these pieces.

1

2

Problem 10There is a triangle on a chessboard (a). One can roll it over any side. Can the triangle occupy the position (b) after any number of moves?

a b

Hint

The long side of the triangle will always go either from the upper left to the lower right corner of a black square, orfrom the upper right to the lower left corner of a white square.

AnswerThe triangle can never occupy position (b). When it gets to the top left square, its long side will always go from the upper right to the lower left corner of the square, as shown below.

or